Tangents

Links and useful resources

• gr10's Geometry >>>START HERE<<<
• AoPS Online Textbook
• AoPS Alcumus
• Big Ideas Geometry textbook
• GeoGebra Online Geometry Constuction Tool
• Two-column math templates
• Proofs unit slides from mathgiraffe.com
• Proofs unit printables for two-column proofs

Tangent Lines

A line is tangent to a circle when it just touches the circle at one point. A tangent line never goes through the interior of a cirlce, because that would require it to cross into, then back out of the circle, making it a secant line instead.

Properties of the tangent line

• If $m$ is tangent to circle $O$ at $B$, then radius $\overline{OB}$ is perpendicular to $m$.
• If $m$ passes through point $B$ on circle $O$ such that $\overline{OB}\perp m$, then $m$ is tangent to circle $O$ at $B$.
• The angle formed by a tangent and a secant is half the difference of the intercepted arcs.

$\mathrm{\angle }BAD=\frac{\text{arc BD}-\text{arc BC}}{2}$

• An angle formed by a tangent and a chord that passes through the point of tangency is one-half the arc intercepted by the angle:

$\mathrm{\angle }ZAB=\frac{\text{arc AB}}{2}$

Media resources

• Youtube search for "Line tangent to circle"

Guided practice

• Prove that the angle of a chord with a tangent that passes through its endpoint is half of the intercepted arc.
• https://artofproblemsolving.com/ebooks/intro-geometry-ebook/par/4317
• https://artofproblemsolving.com/ebooks/intro-geometry-ebook/par/4323

Homework

• #hw (geom-gr10) 12.3 Tangents (p. 316) (gr10) 2025-02-14